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On the Existence of Rainbow 4-Term Arithmetic Progressions

Graphs and Combinatorics volume 23, pages 249–254 (2007)Cite this article

Abstract

For infinitely many natural numbers n, we construct 4-colorings of [n]  =  {1, 2, ..., n}, with equinumerous color classes, that contain no 4-term arithmetic progression whose elements are colored in distinct colors. This result solves an open problem of Jungić et al. (Comb Probab Comput 12:599–620, 2003) Axenovich and Fon-der-Flaass (Electron J Comb 11:R1, 2004).

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References

  • Axenovich, M., Fon-Der-Flaass, D.: On rainbow arithmetic progressions. Electron. J. Comb. 11, R1 (2004)

  • Landman, B., Robertson, A.: Ramsey theory on the integers. American Mathematical Society. Student Mathematical Library 24 (2003)

  • Conlon, D.: Rainbow solutions of linear equations over \({\mathbb{Z}}_{p}\) . Discrete Math. 306, 2056–2063 (2006)

    Google Scholar 

  • Graham, R. L., Rothschild, B. L., Spencer, J. H.: Ramsey theory. Wiley, New York (1990)

  • Jungić, V., Licht (Fox), J., Mahdian, M., Nešetřil, J., Radoičić, R.: Rainbow arithmetic progressions and anti-Ramsey results. Comb. Probab. Comput. 12, 599–620 (2003)

    Google Scholar 

  • Jungić, V., Radoičić, R.: Rainbow 3-term arithmetic progressions. Integers Electron. J. Combin. Number Theory 3, A18 (2003)

  • Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27, 199–245 (1975)

    Google Scholar 

  • van der Waerden, B. L.: Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wis. 15, 212–216 (1927)

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, England

    David Conlon

  2. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Veselin Jungić

  3. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Radoš Radoičić

  4. Baruch College, CUNY, New York, NY, 10010, USA

    Radoš Radoičić

Authors
  1. David Conlon
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  2. Veselin Jungić
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  3. Radoš Radoičić
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Correspondence to Radoš Radoičić.

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Conlon, D., Jungić, V. & Radoičić, R. On the Existence of Rainbow 4-Term Arithmetic Progressions. Graphs and Combinatorics 23, 249–254 (2007). https://doi.org/10.1007/s00373-007-0723-2

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  • DOI: https://doi.org/10.1007/s00373-007-0723-2

Keywords

  • Arithmetic Progression
  • Distinct Color
  • Color Class
  • Acta Arith
  • Consecutive Integer